Embedding of Walsh Brownian Motion

Abstract

Let $(Z,\kappa )$ be a Walsh Brownian motion with spinning measure $\kappa$. Suppose $\mu$ is a probability measure on ${\mathbb{R}}^n$. We characterize all the $\kappa$ such that $\mu$ is a stopping distribution of $(Z,\kappa )$. If we further restrict the solution to be integrable, we show that there would be only one choice of $\kappa$. We also generalize Vallois' embedding, and prove that it minimizes the expectation $\mathbb{E}[\Psi (L_{\tau }^Z)]$ among all the admissible solutions $\tau$, where $\Psi$ is a strictly convex function and $(L_t^Z{)}_{t\ge 0}$ is the local time of the Walsh Brownian motion at the origin.

Keywords: Skorokhod embedding problem, Walsh Brownian motion, Stochastic Calculus, Excursion theory, Vallois' embedding